There are people reining on full blooded andalusians, so why not cut, etc, IMO. Heres a link to the "reining andalusian" Ichibon DMF "Petri" - The Reining Andalusian | Facebook
I do see limitations, like others have stated, if you were planning on showing against open/non pro caliber horses, but if you stay in the lower classes, I bet you'd do fine if you could find one that liked the work! I'd say you are going to have more trouble finding one that would suit, I don't think they are trained for the sport, so you'd probably have to invest from the ground up, training-wise.
I'd say go for it if you were just particularly interested in that breed. If you are just kind of looking around, feeling a couple breeds out, i'd go for a well bred stock type like an appaloosa/paint/quarter that is built for that type of work.